Solving Variational Inequality Problems with Linear Constraints by a Proximal Decomposition Algorithm

The alternating direction method solves large scale variational inequality problems with linear constraints via solving a series of small scale variational inequality problems with simple constraints. The algorithm is attractive if the subproblems can be solved efficiently and exactly. However, the subproblem is itself variational inequality problem, which is structurally also difficult to solve. In this paper, we develop a new decomposition algorithm, which, at each iteration, just solves a system of well-conditioned linear equations and performs a line search. We allow to solve the subproblem approximately and the accuracy criterion is the constructive one developed recently by Solodov and Svaiter. Under mild assumptions on the problem's data, the algorithm is proved to converge globally. Some preliminary computational results are also reported to illustrate the efficiency of the algorithm.     

Convergence Properties of Projection and Contraction Methods for Variational Inequality Problems

In this paper we develop the convergence theory of a general class of projection and contraction algorithms (PC method), where an extended stepsize rule is used, for solving variational inequality (VI) problems. It is shown that, by defining a scaled projection residue, the PC method forces the sequence of the residues to zero. It is also shown that, by defining a projected function, the PC method forces the sequence of projected functions to zero. A consequence of this result is that if the PC method converges to a nondegenerate solution of the VI problem, then after a finite number of iterations, the optimal face is identified. Finally, we study local convergence behavior of the extragradient algorithm for solving the KKT system of the inequality constrained VI problem. \keywords{Variational inequality, Projection and contraction method, Predictor-corrector stepsize, Convergence property.} \amsclass{90C30, 90C33, 65K05.} Accepted 5 September 2000. Online publication 16 January 2001.     

Global Method for Monotone Variational Inequality Problems with Inequality Constraints

We consider optimization methods for monotone variational inequality problems with nonlinear inequality constraints. First, we study the mixed complementarity problem based on the original problem. Then, a merit function for the mixed complementarity problem is proposed, and some desirable properties of the merit function are obtained. Through the merit function, the original variational inequality problem is reformulated as simple bounded minimization. Under certain assumptions, we show that any stationary point of the optimization problem is a solution of the problem considered. Finally, we propose a descent method for the variational inequality problem and prove its global convergence.     

A Simply Constrained Optimization Reformulation of KKT Systems Arising from Variational Inequalities

The Karush—Kuhn—Tucker (KKT) conditions can be regarded as optimality conditions for both variational inequalities and constrained optimization problems. In order to overcome some drawbacks of recently proposed reformulations of KKT systems, we propose casting KKT systems as a minimization problem with nonnegativity constraints on some of the variables. We prove that, under fairly mild assumptions, every stationary point of this constrained minimization problem is a solution of the KKT conditions. Based on this reformulation, a new algorithm for the solution of the KKT conditions is suggested and shown to have some strong global and local convergence properties. Accepted 10 December 1997     

一类单调变分不等式的非精确交替方向法

交替方向法适合于求解大规模问题.该文对于一类变分不等式提出了一种新的交替方向法.在每步迭代计算中,新方法提出了易于计算的子问题,该子问题由强单调的线性变分不等式和良态的非线性方程系统构成.基于子问题的精确求解,该文证明了算法的收敛性.进一步,又提出了一类非精确交替方向法,每步迭代计算只需非精确求解子问题.在一定的非精确条件下,算法的收敛性得以证明.     

Smoothing Functions and Smoothing Newton Method for Complementarity and Variational Inequality Problems

This paper provides for the first time some computable smoothing functions for variational inequality problems with general constraints. This paper proposes also a new version of the smoothing Newton method and establishes its global and superlinear (quadratic) convergence under conditions weaker than those previously used in the literature. These are achieved by introducing a general definition for smoothing functions, which include almost all the existing smoothing functions as special cases.     

Tikhonov Regularization Methods for Variational Inequality Problems

Motivated by the work of Facchinei and Kanzow (Ref. 1) on regularization methods for the nonlinear complementarity problem and the work of Ravindran and Gowda (Ref. 2) for the box variational inequality problem, we study regularization methods for the general variational inequality problem. A sufficient condition is given which guarantees that the union of the solution sets of the regularized problems is nonempty and bounded. It is shown that solutions of the regularized problems form a minimizing sequence of the D-gap function under a mild condition.     

Inexact Operator Splitting Methods with Selfadaptive Strategy for Variational Inequality Problems

The Peaceman-Rachford and Douglas-Rachford operator splitting methods are advantageous for solving variational inequality problems, since they attack the original problems via solving a sequence of systems of smooth equations, which are much easier to solve than the variational inequalities. However, solving the subproblems exactly may be prohibitively difficult or even impossible. In this paper, we propose an inexact operator splitting method, where the subproblems are solved approximately with some relative error tolerance. Another contribution is that we adjust the scalar parameter automatically at each iteration and the adjustment parameter can be a positive constant, which makes the methods more practical and efficient. We prove the convergence of the method and present some preliminary computational results, showing that the proposed method is promising. This work was supported by the NSFC grant 10501024.     

一类变分不等式问题的信赖域算法

基于J.M.Peng研究一类变分不等式问题（简记为VIP）时所提出的价值函数，本文提出了求解强单调的VIP的一个新的信赖域算法。和已有的处理VIP的信赖域方法不同的是：它在每步迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。这样，计算的复杂性一般来说可降低。在通常的假设条件下，文中还证明了算法的整体收敛性。最后，在梯度是半光滑和约束是矩形域的假设下，该算法还是超线性收敛的。     

New Alternating Direction Method for a Class of Nonlinear Variational Inequality Problems

The alternating direction method is an attractive method for a class of variational inequality problems if the subproblems can be solved efficiently. However, solving the subproblems exactly is expensive even when the subproblem is strongly monotone or linear. To overcome this disadvantage, this paper develops a new alternating direction method for cocoercive nonlinear variational inequality problems. To illustrate the performance of this approach, we implement it for traffic assignment problems with fixed demand and for large-scale spatial price equilibrium problems.     

Decomposition Method for a Class of Monotone Variational Inequality Problems

In the solution of the monotone variational inequality problem VI(, F), with

一类非对称单调变分不等式的交替方向法

对一类非对称变分不等式问题提出了交替方向法。推广了交替方向仅适用于等式约束或不等约束的情形，得出了迭代序列的一些性质及收敛性．     

Equivalent Optimization Formulations and Error Bounds for Variational Inequality Problems

We investigate whether some merit functions for variational inequality problems (VIP) provide error bounds for the underlying VIP. Under the condition that the involved mapping F is strongly monotone, but not necessarily Lipschitz continuous, we prove that the so-called regularized gap function provides an error bound for the underlying VIP. We give also an example showing that the so-called D-gap function might not provide error bounds for a strongly monotone VIP.This research was supported by United College and by a direct grant of the Chinese University of Hong Kong. The authors thank the referees for helpful comments and suggestions.     

A General Approach to Convergence Properties of Some Methods for Nonsmooth Convex Optimization

Based on the notion of the ε -subgradient, we present a unified technique to establish convergence properties of several methods for nonsmooth convex minimization problems. Starting from the technical results, we obtain the global convergence of: (i) the variable metric proximal methods presented by Bonnans, Gilbert, Lemaréchal, and Sagastizábal, (ii) some algorithms proposed by Correa and Lemaréchal, and (iii) the proximal point algorithm given by Rockafellar. In particular, we prove that the Rockafellar—Todd phenomenon does not occur for each of the above mentioned methods. Moreover, we explore the convergence rate of {||x _k || } and {f(x _k ) } when {x _k } is unbounded and {f(x _k ) } is bounded for the non\-smooth minimization methods (i), (ii), and (iii). Accepted 15 October 1996     

Alternating Direction Method for a Class of Sylvester Matrix Equations with Linear Matrix Inequality Constraint

This paper introduces an alternating direction method of multipliers (ADMM) for finding solutions to a class of Sylvester matrix equation AXB = E subject to a linear matrix inequality constraint CXD≥G. Preliminary convergence properties of ADMM are presented. Numerical experiments are performed to illustrate the feasibility and effectiveness of ADMM. In addition, some numerical comparisons with a recent algorithm are also given.     

Quadratic Convergence of a Long-Step Interior-Point Method for Nonlinear Monotone Variational Inequality Problems

This paper offers an analysis on a standard long-step primal-dual interior-point method for nonlinear monotone variational inequality problems. The method has polynomial-time complexity and its q-order of convergence is two. The results are proved under mild assumptions. In particular, new conditions on the invariance of the rank and range space of certain matrices are employed, rather than restrictive assumptions like nondegeneracy.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Non-Interior Continuation Method for Solving the Monotone Semidefinite Complementarity Problem

Recently, Chen and Tseng extended non-interior continuation/ smooth- ing methods for solving linear/ nonlinear complementarity problems to semidefinite complementarity problems (SDCP). In this paper we propose a non-interior continuation method for solving the monotone SDCP based on the smoothed Fischer—Burmeister function, which is shown to be globally linearly and locally quadratically convergent under suitable assumptions. Our algorithm needs at most to solve a linear system of equations at each iteration. In addition, in our analysis on global linear convergence of the algorithm, we need not use the assumption that the Fréchet derivative of the function involved in the SDCP is Lipschitz continuous. For non-interior continuation/ smoothing methods for solving the nonlinear complementarity problem, such an assumption has been used widely in the literature in order to achieve global linear convergence results of the algorithms.     

Unconstrained Optimization Reformulations of Variational Inequality Problems

Recently, Peng considered a merit function for the variational inequality problem (VIP), which constitutes an unconstrained differentiable optimization reformulation of VIP. In this paper, we generalize the merit function proposed by Peng and study various properties of the generalized function. We call this function the D-gap function. We give conditions under which any stationary point of the D-gap function is a solution of VIP and conditions under which it provides a global error bound for VIP. We also present a descent method for solving VIP based on the D-gap function.     

A Regularization Newton Method for Solving Nonlinear Complementarity Problems

In this paper we construct a regularization Newton method for solving the nonlinear complementarity problem (NCP(F )) and analyze its convergence properties under the assumption that F is a P ₀ -function. We prove that every accumulation point of the sequence of iterates is a solution of NCP(F ) and that the sequence of iterates is bounded if the solution set of NCP(F ) is nonempty and bounded. Moreover, if F is a monotone and Lipschitz continuous function, we prove that the sequence of iterates is bounded if and only if the solution set of NCP(F ) is nonempty by setting , where is a parameter. If NCP(F) has a locally unique solution and satisfies a nonsingularity condition, then the convergence rate is superlinear (quadratic) without strict complementarity conditions. At each step, we only solve a linear system of equations. Numerical results are provided and further applications to other problems are discussed. Accepted 25 March 1998